Polygonal Numbers

A polygonal number is defined as “A type of figurate number which is a generalization of triangular, square, etc., numbers to an arbitrary n-gonal number. The above diagrams graphically illustrate the process by which the polygonal numbers are built up.” (Mathworld.wolfram.com) Every student of school mathematics knows about the square numbers, and many know about the triangular numbers as well. But less familiar are the pentagonal, hexagonal, etc. varieties.

Even less well known is the fact that each of those types of numbers has a cousin of sorts, called the centered polygonal numbers. Yes, the regular triangular numbers have their corresponding “centered” form. The same is true for the squares, pentagonals, hexagonals, etc. (See diagram below.)

Therefore, our definition for these numbers is “A figurate number in which layers of polygons are drawn centered about a point instead of with the point at a vertex.” (Mathworld.wolfram.com)

Many facts and theorems are known about polygonal numbers, especially of the squares and triangulars. We wouldn’t be able to talk about the Pythagorean Theorem if it weren’t for the squares, just to mention the most famous example of all. And the triangulars arise whenever we are concerned with the sum of consecutive integers, from 1 to n.

What I want to do in this page of WTM is present some ideas that are not normally covered in an average school math class, yet ideas that are well within the understanding of most students. First, we will show the algebraic formulas for both the regular and centered polygonal numbers, up to a level seldom discussed: a 30-sided polygon!

The Formulas

Number
of Sides

Regular
form

Centered
form

3

n(n + 1)/2

(3n2 – 3n + 2)/2

4

n2

2n2 – 2n + 1

5

n(3n – 1)/2

(5n2 – 5n + 2)/2

6

n(2n – 1)

3n2 – 3n + 1

7

n(5n – 3)/2

(7n2 – 7n + 2)/2

8

n(3n – 2)

4n2 – 4n + 1

9

n(7n – 5)/2

(9n2 – 9n + 2)/2

10

n(4n – 3)

5n2 – 5n + 1

11

n(9n – 7)/2

(11n2 – 11n + 2)/2

12

n(5n – 4)

6n2 – 6n + 1

13

n(11n – 9)/2

(13n2 – 13n + 2)/2

14

n(6n – 5)

7n2 – 7n + 1

15

n(13n – 11)/2

(15n2 – 15n + 2)/2

16

n(7n – 6)

8n2 – 8n + 1

17

n(15n – 13)/2

(17n2 – 17n + 2)/2

18

n(8n – 7)

9n2 – 9n + 1

19

n(17n – 15)/2

(19n2 – 19n + 2)/2

20

n(9n – 8)

10n2 – 10n + 1

21

n(19n – 17)/2

(21n2 – 21n + 2)/2

22

n(10n – 9)

11n2 – 11n + 1

23

n(21n – 19)/2

(23n2 – 23n + 2)/2

24

n(11n – 10)

12n2 – 12n + 1

25

n(23n – 21)/2

(25n2 – 25n + 2)/2

26

n(12n – 11)

13n2 – 13n + 1

27

n(25n – 23)/2

(27n2 – 27n + 2)/2

28

n(13n – 12)/2

14n2 – 14n + 1

29

n(27n – 25)/2

(29n2 – 29n + 2)/2

30

n(14n – 13)

15n2 – 15n + 1

Hey! Do you see a pattern in the table? If you do, perhaps you could write a general formula for it; then you could give the formula for any n-gonal number of either type, without using the table, and even beyond 30 sides.

And now for some numbers…

Our next chart will give us some actual numbers for the polygons up to decagons.

The First Nine Terms

Name

Regular

Centered

triangular

1, 3, 6, 10, 15, 21, 28, 36, 45, …

1, 4, 10, 19, 31, 46, 64, 85, 109, …

square

1, 4, 9, 16, 25, 36, 49, 64, 81, …

1, 5, 13, 25, 41, 61, 85, 113, 145, …

pentagonal

1, 5, 12, 22, 35, 51, 70, 92, 117, …

1, 6, 16, 31, 51, 76, 106, 141, 181, …

hexagonal

1, 6, 15, 28, 45, 66, 91, 120, 153, …

1, 7, 19, 37, 61, 91, 127, 169, 217, …

heptagonal

1, 7, 18, 34, 55, 81, 112, 148, 189

1, 8, 22, 43, 71, 106, 148, 197, 253, …

octagonal

1, 8, 21, 40, 65, 96, 133, 176, 225, …

1, 9, 25, 49, 81, 121, 169, 225, 289, …

nonagonal

1, 9, 24, 46, 75, 111, 154, 204, 261, …

1, 10, 28, 55, 91, 136, 190, 253, 325, …

decagonal

1, 10, 27, 52, 85, 126, 175, 232, 297, …

1, 11, 31, 60, 101, 151, 211, 281, 361, …

Now that we have some numbers, what should we do with them? If I may paraphrase an old popular song by Nancy Sinatra, these numbers are made for adding! So consider this…

We again turn to Mathworld for some vital information: Fermat’s Polygonal Number Theorem
In 1638, Fermat proposed that every positive integer is a sum of at most three triangular numbers, four square numbers, five pentagonal numbers, and n n-polygonal numbers. Fermat claimed to have a proof of this result, although Fermat’s proof has never been found. Gauss proved the triangular case, and noted the event in his diary on July 10, 1796, with the notation

What that little cryptic notation means is that all whole numbers can be expressed as the sum of three, or fewer, triangular numbers. Here is an interesting example:

100 = 91 + 6 + 3 = T13 + T3 + T2

100 = 55 + 45 = T10 + T9

This illustrates that sometimes a number has two possibilities, with 3 or 2 terms. Nice, huh?

Turning now to the case of the squares… Fermat said that all whole numbers can be expressed as the sum of four, or fewer, square numbers. Let’s look at this example:

50 = 49 + 1 = S7 + S1

50 = 25 + 25 = S5 + S5

50 = 25 + 16 + 9 = S5 + S4 + S3

50 = 36 + 9 + 4 + 1 = S6 + S3 + S2 + S1

Notice that there were expressions with 2, 3, and 4 terms. Thereby, arises an interesting idea: given a particular number, how many different expressions can be found? I challenge you to research this and report back to me. Ok?

One more time… For the case of the pentagonals, we can use up to five of them to express all whole numbers. Let’s check out the situation for the number 2002.

2002 = 1001 + 1001 = P26 + P26

2002 = 1520 + 477 + 5 = P32 + P18 + P2

2002 = 1820 + 176 + 5 + 1 = P35 + P11 + P2 + P1

2002 = 1717 + 176 + 92 + 12 + 5 = P34 + P11 + P8 + P3 + P2

As before, we have demonstrated that we can achieve our goal with 2-5 terms. In fact, there are many more such ways to do it than presented here. Finding all possible ways is now more difficult, (unless one uses a computer program).

The Other Side of the Story

So far we have only been working with the regular polygonal numbers. Let’s now look at the centered case. The natural question to ask should be: does there exist an analogue to Fermat’s theorem, as discussed above? Specifically, are three CTN’s (Centered Triangular Numbers) sufficient to express all whole numbers?

The best way to answer that is to start small and work your way up. Here is a chart for the numbers from 1 to 10. Recall, the set of CTN’s is {1, 4, 10, 19, …}.

The CTN Analogue

No.

expression

No.

expression

1

1

6

4 + 1 + 1

2

1 + 1

7

none

3

1 + 1 + 1

8

4 + 4

4

4

9

4 + 4 + 1

5

4 + 1

10

10

Well, I guess that about answers our question, doesn’t it? And it didn’t take long either.

However, it brings to mind yet another question — what is the next impossible number?

[Remember: you can use up to 4 CSN’s and 5 CPN’s, and so on, in the expressons.]

Special Numbers

Another popular activity when one is faced with a long list of numbers is to search for the presence of numbers with special characteristics, such as squares, cubes, or palindromes. Let’s first consider the modern favorite of many mathematicians: palindromes.

The “mother of all websites” dealing with palindromes undoubtedly is World!Of Numbers, edited by Patrick De Geest. In his site you can find an extensive treatment of palindromes that are also triangular numbers, and squares as well. In fact, he gives data for the pentagonals up to the nonagonals. We heartily encourage you to visit that site; you will be justly rewarded for your time and efforts.

However, all the data to be found there uses only the regular type of polygonals; there is nothing mentioned about the centered type. Here is our attempt to fill in that gap of trivia information. (Note: At present, our search only shows results up to n = 300 and for k-gonals from k = 3 to 40. We also omit any single-digit palindromes as being trivial in this context.)

The Palindromes

k

n

CPkN(n)

Prime Factorization

3

101

15151

109 x 139

174

45154

2 x 107 x 211

211

66466

2 x 167 x 199

249

92629

211 x 439

257

98689

prime

4

10

181

prime

13

313

prime

17

545

5 x 109

5

8

141

3 x 47

9

181

prime

65

10401

3 x 3467

6

18

919

prime

7

3

22

2 x 11

20

1331

113

8

6

121

112

51

10201

1012

56

12321

32 x 372

61

14641

114

9

4

55

5 x 11

12

595

5 x 7 x 17

10

2

11

prime

5

101

prime

11

5

111

3 x 37

7

232

23 x 29

11

606

2 x 3 x 101

12

727

prime

62

20802

2 x 3 x 3467

12

5

121

112

6

181

prime

16

1441

11 x 131

46

12421

prime

13

5

131

prime

14

5

141

3 x 47

8

393

3 x 131

9

505

5 x 101

15

5

151

prime

10

676

22 x 132

52

19891

prime

16

5

161

7 x 23

46

16561

prime

17

5

171

32 x 19

93

72727

prime

18

3

55

5 x 11

5

181

prime

8

505

5 x 101

40

14041

19 x 739

19

5

191

prime

20

4

121

112

21

2

22

2 x 11

9

757

prime

36

13231

101 x 131

120

149941

11 x 43 x 317

255

680086

2 x 11 x 19 x 1627

22

none

in this

range yet

23

7

484

22 x 112

29

9339

3 x 11 x 283

24

7

505

5 x 101

25

4

151

prime

36

15751

19 x 829

289

1040401

101 x 10301

26

30

11311

prime

27

8

757

prime

28

35

16661

prime

29

3

88

23 x 11

30

4

181

prime

94

131131

7 x 11 x 13 x 131

260

1010101

73 x 101 x 137

31

69

72727

prime

32

2

33

3 x 11

10

1441

11 x 131

26

10401

19 x 739

251

1004001

3 x 334667

33

260

1111111

239 x 4649

34

25

10201

1012

43

30703

prime

172

500005

5 x 11 x 9091

35

25

10501

prime

28

13231

101 x 131

33

18481

prime

36

25

10801

7 x 1543

260

1212121

prime

37

none

in this

range yet

38

31

17671

41 x 431

39

52

51715

5 x 10343

260

1313131

17 x 77243

40

3

121

112

9

1441

11 x 131

27

14041

19 x 739

225

1008001

prime

While studying the results above, I saw two rather interesting numbers: 1212121 and 1313131. Not only do they share an obvious digital pattern of 1d1d1d1, but they both are the 260th term in their respective orders (k = 36 and 39).

Now if we look back at the 33-gonal and 30-gonal lists, we see 1111111 and 1010101. As one might begin to suspect by now, they are the 260th terms there. So it’s table time again!

The 260th Term Case

k

Number

Prime Factorization

30

1010101

73 x 101 x 137

33

1111111

239 x4649

36

1212121

prime

39

1313131

17 x 77243

42

1414141

43 x 32887

45

1515151

11 x 181 x 761

48

1616161

prime

51

1717171

199 x 8629

54

1818181

31 x 89 x 659

57

1919191

29 x 66179

Stop the Presses!!! (4/22/2002)

Just in to the editorial offices of WTM! Patrick De Geest has sent in a pair of 6-packs…of palindromes for the missing data in the chart of Palindromes above. Here it is.

Palindromes for k = 22 & 37

k

n

CPkN(n)

Prime Factorization

22

4156

189949981

13 x 14611537

524962

3031430341303

7 x 13 x 33312421333

321895111

1139781083801879311

13 x 53 x 163 x 15259 x 665102447

358542860

1414082803082804141

7 x 19 x 67 x 349 x 2671 x 170235089

362349816

1444271276721724441

83 x 17400858755683427

422820435

1966548318138456691

17 x 191 x 5346613 x 113277481

37

378

2636362

2 x 163 x 8087

2400

106515601

43 x 2477107

407157

3066863686603

prime

2835585

148749979947841

859 x 56099 x 3086801

3443283

219339595933912

23 x 27417449491739

6792834

853637858736358

2 x 17 x 25106995845187

Also, my friend and colleague from Romania, Andrei Lazanu, sent along some important data regarding the matter above about the impossible cases for numbers to be expressed as the sum of 3, or fewer, CTN’s. According to the program he wrote, there are over 70 numbers less than 200 that can not be decomposed in this way. How many can you find? Can you find all of them?

Andrei also was kind enough to provide WTM with some information about how many ways 2002 can be expressed with 5, or fewer, regular pentagonal numbers. It can be done in 166 ways. [See examples above.]

In addition, he informs us that the same task can be achieved in 31 ways with regular triangular numbers, and in 101 ways using regular square numbers. (Thanks, Andrei.)

Update (5/13/02)

De Geest has provided WTM with some more CTN palindromes. Here they are (as of 4/25/02).

More CTN Palindromes
(n < 3,711,895,911)

n

CTN

Prime Factorization

920

1690961

29 * 58309

1258

3162613

101 * 173 * 181

1263

3187813

prime

1622

5258525

52 * 17 * 12373

1707

5824285

5 * 17 * 68521

170707

58281418285

5 * 39821 * 292717

904281

1635446445361

109 * 9461 * 1585889

1258183

3166046406613

173 * 13537 * 1351913

7901015

124852060258421

21589 * 5783133089

8659930

149988757889941

17 * 852013 * 10355321

12458598

310433303334013

101 * 3073597062713

17070707

582818040818285

5 * 89 * 461693 * 2836741

80472265

12951570707515921

13 * 17 * 113 * 1609 * 322326253

1616689804

5227371841481737225

52 * 941 * 104917 * 2117911937

1680689789

5649436330336349465

5 * 29 * 761 * 51197936746897

1705387644

5816694029204966185

5 * 1163338805840993237

Next (5/13/02), Andrei has extended the matter of Centered Hexagonal Numbers (CP6N) if ever so slightly…

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One thought on “Polygonal Numbers”

The list of indices of palindromic centered triangular numbers differs from the one
in the OEIS, http://oeis.org/A005448 . 3*n(n-1)/2+1 is palindromic for n = 1, 2, 101, 174, 211, 249, 257, 1822, 2070, 20795,… The update of 5/13/02 does not
fit in there.
Richard